Question

Use the properties of exponents and radicals to generate an equivalent expression for (256√10)^5/512^1/3.

Answer 1

To simplify the expression \((256 \sqrt{10})^5 / 512^{1/3}\), we can start by breaking down the components step by step.

1. **Simplify \( (256 \sqrt{10})^5 \)**:

- \(256\) can be expressed as \(2^8\) since \(256 = 2^8\).
- The square root of \(10\) can be expressed as \(10^{1/2}\).
- Therefore, we rewrite:
\[
256 \sqrt{10} = 2^8 \cdot 10^{1/2}
\]
- Now we apply the exponent:
\[
(256 \sqrt{10})^5 = (2^8 \cdot 10^{1/2})^5 = 2^{8 \cdot 5} \cdot 10^{(1/2) \cdot 5} = 2^{40} \cdot 10^{5/2}
\]

2. **Simplify \(512^{1/3}\)**:

- \(512\) can be expressed as \(2^9\) since \(512 = 2^9\).
- Now we can simplify:
\[
512^{1/3} = (2^9)^{1/3} = 2^{9/3} = 2^{3} = 8
\]

3. **Rewrite the original expression using the simplified forms**:
\[
\frac{(256 \sqrt{10})^5}{512^{1/3}} = \frac{2^{40} \cdot 10^{5/2}}{2^{3}}
\]

4. **Combine the exponents of \(2\)**:

- We subtract the exponents in the numerator and the denominator:
\[
2^{40} / 2^{3} = 2^{40 - 3} = 2^{37}
\]
- Thus, we have:
\[
\frac{(256 \sqrt{10})^5}{512^{1/3}} = 2^{37} \cdot 10^{5/2}
\]

5. **Express \(10^{5/2}\) in terms of radicals**:

- We can rewrite \(10^{5/2}\) as:
\[
10^{5/2} = (10^5)^{1/2} = \sqrt{10^5} = \sqrt{10^4 \cdot 10} = 10^2 \cdot \sqrt{10} = 100 \sqrt{10}
\]

6. **Final expression**:

- Therefore, the entire expression can be combined and simplified to:
\[
2^{37} \cdot 10^{5/2} = 2^{37} \cdot 100 \sqrt{10}
\]

So the equivalent expression is:

\[
\boxed{2^{37} \cdot 100 \sqrt{10}}
\]